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<H1><A NAME="SECTION034100000000000000000"></A><A NAME="secgendef"></A>
<BR>
Error Bounds for the Generalized Symmetric Definite
Eigenproblem
</H1>

<P>
There are three types of problems to consider.
<A NAME="11841"></A>
In all cases <B><I>A</I></B> and <B><I>B</I></B>
are real symmetric (or complex Hermitian) and <B><I>B</I></B> is positive definite.
These decompositions are computed for real symmetric matrices
by the driver routines
xSYGV, xSYGVX, xSYGVD,
xSPGV, xSPGVX, xSPGVD,
and (for type 1 only)
xSBGV, xSBGVX and xSBGVD
(see subsection <A HREF="node34.html#secGSEP">2.3.5.1</A>).
These decompositions are computed for complex Hermitian matrices
by the driver routines
xHEGV, xHEGVX, xHEGVD,
xHPGV, xHPGVX, xHPGVD,
and (for type 1 only)
xHBGV, xHBGVX, xHBGVD
(see subsection <A HREF="node34.html#secGSEP">2.3.5.1</A>).
In each of the following three decompositions,
<IMG
 WIDTH="16" HEIGHT="15" ALIGN="BOTTOM" BORDER="0"
 SRC="img28.gif"
 ALT="$\Lambda$">
is real and diagonal with diagonal entries

<!-- MATH
 $\lambda_1 \leq \cdots \leq \lambda_n$
 -->
<IMG
 WIDTH="108" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
 SRC="img579.gif"
 ALT="$\lambda_1 \leq \cdots \leq \lambda_n$">,
and
the columns <B><I>z</I><SUB><I>i</I></SUB></B> of <B><I>Z</I></B> are linearly independent vectors.
The <IMG
 WIDTH="20" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
 SRC="img523.gif"
 ALT="$\lambda_i$">
are called
<B>eigenvalues</B> and the <B><I>z</I><SUB><I>i</I></SUB></B> are
<B>eigenvectors</B>.<A NAME="11846"></A><A NAME="11847"></A>
<A NAME="11848"></A> <A NAME="11849"></A> <A NAME="11850"></A> <A NAME="11851"></A> <A NAME="11852"></A> <A NAME="11853"></A>
<A NAME="11854"></A><A NAME="11855"></A><A NAME="11856"></A><A NAME="11857"></A><A NAME="11858"></A><A NAME="11859"></A>
<A NAME="11860"></A><A NAME="11861"></A><A NAME="11862"></A><A NAME="11863"></A><A NAME="11864"></A><A NAME="11865"></A>
<A NAME="11866"></A> <A NAME="11867"></A> <A NAME="11868"></A> <A NAME="11869"></A> <A NAME="11870"></A> <A NAME="11871"></A>
<A NAME="11872"></A><A NAME="11873"></A><A NAME="11874"></A><A NAME="11875"></A><A NAME="11876"></A><A NAME="11877"></A>
<A NAME="11878"></A><A NAME="11879"></A><A NAME="11880"></A><A NAME="11881"></A><A NAME="11882"></A><A NAME="11883"></A>

<P>
<DL COMPACT>
<DT>1.
<DD><IMG
 WIDTH="63" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
 SRC="img56.gif"
 ALT="$A - \lambda B$">.
The eigendecomposition may be written 
<!-- MATH
 $Z^T A Z = \Lambda$
 -->
<IMG
 WIDTH="90" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
 SRC="img42.gif"
 ALT="$Z^T A Z = \Lambda$">
and
<B><I>Z</I><SUP><I>T</I></SUP> <I>B Z</I> = <I>I</I></B> (or 
<!-- MATH
 $Z^H A Z = \Lambda$
 -->
<IMG
 WIDTH="92" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
 SRC="img671.gif"
 ALT="$Z^H A Z = \Lambda$">
and <B><I>Z</I><SUP><I>H</I></SUP> <I>B Z</I> = <I>I</I></B>
if <B><I>A</I></B> and <B><I>B</I></B> are complex).
This may also be written 
<!-- MATH
 $Az_i = \lambda_i B z_i$
 -->
<IMG
 WIDTH="97" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
 SRC="img672.gif"
 ALT="$Az_i = \lambda_i B z_i$">.
<P>
<DT>2.
<DD>
<!-- MATH
 $AB - \lambda I$
 -->
<IMG
 WIDTH="72" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
 SRC="img383.gif"
 ALT="$AB- \lambda I$">.
The eigendecomposition may be written 
<!-- MATH
 $Z^{-1} A Z^{-T} = \Lambda$
 -->
<IMG
 WIDTH="118" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
 SRC="img673.gif"
 ALT="$Z^{-1} A Z^{-T} = \Lambda$">
and
<B><I>Z</I><SUP><I>T</I></SUP> <I>B Z</I> = <I>I</I></B> (
<!-- MATH
 $Z^{-1} A Z^{-H} = \Lambda$
 -->
<IMG
 WIDTH="121" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
 SRC="img674.gif"
 ALT="$Z^{-1} A Z^{-H} = \Lambda$">
and <B><I>Z</I><SUP><I>H</I></SUP> <I>B Z</I> = <I>I</I></B> if <B><I>A</I></B>
and <B><I>B</I></B> are complex).
This may also be written 
<!-- MATH
 $ABz_i = \lambda_i z_i$
 -->
<IMG
 WIDTH="97" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
 SRC="img675.gif"
 ALT="$ABz_i = \lambda_i z_i$">.

<P>
<DT>3.
<DD>
<!-- MATH
 $BA - \lambda I$
 -->
<IMG
 WIDTH="72" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
 SRC="img384.gif"
 ALT="$BA- \lambda I$">.
The eigendecomposition may be written 
<!-- MATH
 $Z^T A Z = \Lambda$
 -->
<IMG
 WIDTH="90" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
 SRC="img42.gif"
 ALT="$Z^T A Z = \Lambda$">
and 
<!-- MATH
 $Z^T B^{-1} Z = I$
 -->
<B><I>Z</I><SUP><I>T</I></SUP> <I>B</I><SUP>-1</SUP> <I>Z</I> = <I>I</I></B> (
<!-- MATH
 $Z^H A Z = \Lambda$
 -->
<IMG
 WIDTH="92" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
 SRC="img671.gif"
 ALT="$Z^H A Z = \Lambda$">
and 
<!-- MATH
 $Z^H B^{-1} Z = I$
 -->
<B><I>Z</I><SUP><I>H</I></SUP> <I>B</I><SUP>-1</SUP> <I>Z</I> = <I>I</I></B> if <B><I>A</I></B>
and <B><I>B</I></B> are complex).
This may also be written 
<!-- MATH
 $BAz_i = \lambda_i z_i$
 -->
<IMG
 WIDTH="97" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
 SRC="img676.gif"
 ALT="$BAz_i = \lambda_i z_i$">.

<P>
</DL>

<P>
The approximate error bounds<A NAME="footfnm 0"><SUP>4.10</SUP></A>for the computed eigenvalues 
<!-- MATH
 $\hat{\lambda}_1 \leq \cdots \leq \hat{\lambda}_n$
 -->
<IMG
 WIDTH="108" HEIGHT="41" ALIGN="MIDDLE" BORDER="0"
 SRC="img525.gif"
 ALT="$\hat{\lambda}_1 \leq \cdots \leq \hat{\lambda}_n$">
are
<BR><P></P>
<DIV ALIGN="CENTER">

<!-- MATH
 \begin{displaymath}
| \hat{\lambda}_i - \lambda_i | \leq {\tt EERRBD}(i) \; .
\end{displaymath}
 -->


<IMG
 WIDTH="167" HEIGHT="31" BORDER="0"
 SRC="img677.gif"
 ALT="\begin{displaymath}
\vert \hat{\lambda}_i - \lambda_i \vert \leq {\tt EERRBD}(i) \; .
\end{displaymath}">
</DIV>
<BR CLEAR="ALL">
<P></P>
The approximate error
bounds<A NAME="11897"></A><A NAME="11898"></A>
for the computed eigenvectors <IMG
 WIDTH="18" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
 SRC="img527.gif"
 ALT="$\hat{z}_i$">,
which bound the acute angles between the computed eigenvectors and true
eigenvectors <B><I>z</I><SUB><I>i</I></SUB></B>, are
<A NAME="11900"></A>
<A NAME="11901"></A>
<BR><P></P>
<DIV ALIGN="CENTER">

<!-- MATH
 \begin{displaymath}
\theta ( \hat{z}_i , z_i ) \leq {\tt ZERRBD}(i) \; .
\end{displaymath}
 -->


<IMG
 WIDTH="164" HEIGHT="31" BORDER="0"
 SRC="img528.gif"
 ALT="\begin{displaymath}
\theta ( \hat{z}_i , z_i ) \leq {\tt ZERRBD} (i) \; .
\end{displaymath}">
</DIV>
<BR CLEAR="ALL">
<P></P>
These bounds are computed differently, depending on which of the above three
problems are to be solved. The following code fragments show how.

<P>
<DL COMPACT>
<DT>1.
<DD>First we consider error bounds for problem 1.
<P>
 
<P>
<PRE>
      EPSMCH = SLAMCH( 'E' )
*     Solve the eigenproblem A - lambda B (ITYPE = 1)
      ITYPE = 1
*     Compute the norms of A and B
      ANORM = SLANSY( '1', UPLO, N, A, LDA, WORK )
      BNORM = SLANSY( '1', UPLO, N, B, LDB, WORK )
*     The eigenvalues are returned in W
*     The eigenvectors are returned in A
      CALL SSYGV( ITYPE, 'V', UPLO, N, A, LDA, B, LDB, W, WORK,
     $            LWORK, INFO )
      IF( INFO.GT.0 .AND. INFO.LE.N ) THEN
         PRINT *,'SSYGV did not converge'
      ELSE IF( INFO.GT.N ) THEN
         PRINT *,'B not positive definite'
      ELSE IF ( N.GT.0 ) THEN
*        Get reciprocal condition number RCONDB of Cholesky factor of B
         CALL STRCON( '1', UPLO, 'N', N, B, LDB, RCONDB, WORK, IWORK,
     $                INFO )
         RCONDB = MAX( RCONDB, EPSMCH )
         CALL SDISNA( 'Eigenvectors', N, N, W, RCONDZ, INFO )
         DO 10 I = 1, N
            EERRBD( I ) = ( EPSMCH / RCONDB**2 ) * ( ANORM / BNORM +
     $                      ABS( W(I) ) )
            ZERRBD( I ) = ( EPSMCH / RCONDB**3 ) * ( ( ANORM / BNORM )
     $                     / RCONDZ(I) + ( ABS( W(I) ) / RCONDZ(I) ) *
     $                     RCONDB )
10       CONTINUE
      END IF
</PRE>
<A NAME="11908"></A>

<P>
For example<A NAME="footfnm 0"><SUP>4.11</SUP></A>, if

<!-- MATH
 ${\tt SLAMCH('E')} = 2^{-24} = 5.961 \cdot 10^{-8}$
 -->
<IMG
 WIDTH="259" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="img397.gif"
 ALT="${\tt SLAMCH('E')} = 2^{-24} = 5.961 \cdot 10^{-8}$">,
<BR><P></P>
<DIV ALIGN="CENTER">

<!-- MATH
 \begin{displaymath}
A = \left( \begin{array}{ccc} 100000 & 10099 & 2109 \\10099 & 100020 & 10012 \\
2109 & 10112 & -48461 \end{array} \right) 
\; \; {\rm and} \; \;
B = \left( \begin{array}{ccc} 99 & 10 & 2 \\10 & 100 & 10 \\2 & 10 & 100 \end{array} \right)
\end{displaymath}
 -->


<IMG
 WIDTH="503" HEIGHT="73" BORDER="0"
 SRC="img678.gif"
 ALT="\begin{displaymath}
A = \left( \begin{array}{ccc} 100000 &amp; 10099 &amp; 2109 \\ 10099...
...&amp; 10 &amp; 2 \\ 10 &amp; 100 &amp; 10 \\ 2 &amp; 10 &amp; 100 \end{array} \right)
\end{displaymath}">
</DIV>
<BR CLEAR="ALL">
<P></P>
then 
<!-- MATH
 ${\tt ANORM} = 120231$
 -->
<IMG
 WIDTH="126" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="img679.gif"
 ALT="${\tt ANORM} = 120231$">,

<!-- MATH
 ${\tt BNORM} = 120$
 -->
<IMG
 WIDTH="99" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="img680.gif"
 ALT="${\tt BNORM} = 120$">,
and

<!-- MATH
 ${\tt RCONDB} = .8326$
 -->
<IMG
 WIDTH="122" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="img681.gif"
 ALT="${\tt RCONDB} = .8326$">,
and
the approximate eigenvalues, approximate error bounds,
and true errors are

<P>
<DIV ALIGN="CENTER">
<TABLE CELLPADDING=3 BORDER="1">
<TR><TD ALIGN="CENTER"><B><I>i</I></B></TD>
<TD ALIGN="CENTER"><IMG
 WIDTH="20" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
 SRC="img523.gif"
 ALT="$\lambda_i$"></TD>
<TD ALIGN="CENTER"><TT> EERRBD</TT><B>(<I>i</I>)</B></TD>
<TD ALIGN="CENTER">true 
<!-- MATH
 $| \hat{\lambda}_i - \lambda_i |$
 -->
<IMG
 WIDTH="67" HEIGHT="41" ALIGN="MIDDLE" BORDER="0"
 SRC="img531.gif"
 ALT="$\vert \hat{\lambda}_i - \lambda_i \vert$"></TD>
<TD ALIGN="CENTER"><TT> ZERRBD</TT><B>(<I>i</I>)</B></TD>
<TD ALIGN="CENTER">true 
<!-- MATH
 $\theta ( \hat{v}_i , v_i )$
 -->
<IMG
 WIDTH="62" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="img568.gif"
 ALT="$\theta ( \hat{v}_i , v_i )$"></TD>
</TR>
<TR><TD ALIGN="CENTER">1</TD>
<TD ALIGN="CENTER"><B>-500.0</B></TD>
<TD ALIGN="CENTER">
<!-- MATH
 $1.3 \cdot 10^{-4}$
 -->
<IMG
 WIDTH="75" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
 SRC="img682.gif"
 ALT="$1.3 \cdot 10^{-4}$"></TD>
<TD ALIGN="CENTER">
<!-- MATH
 $9.0 \cdot 10^{-6}$
 -->
<IMG
 WIDTH="75" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
 SRC="img683.gif"
 ALT="$9.0 \cdot 10^{-6}$"></TD>
<TD ALIGN="CENTER">
<!-- MATH
 $9.8 \cdot 10^{-8}$
 -->
<IMG
 WIDTH="75" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
 SRC="img684.gif"
 ALT="$9.8 \cdot 10^{-8}$"></TD>
<TD ALIGN="CENTER">
<!-- MATH
 $1.0 \cdot 10^{-9}$
 -->
<IMG
 WIDTH="75" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
 SRC="img685.gif"
 ALT="$1.0 \cdot 10^{-9}$"></TD>
</TR>
<TR><TD ALIGN="CENTER">2</TD>
<TD ALIGN="CENTER"><B>1000.</B></TD>
<TD ALIGN="CENTER">
<!-- MATH
 $1.7 \cdot 10^{-4}$
 -->
<IMG
 WIDTH="75" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
 SRC="img686.gif"
 ALT="$1.7 \cdot 10^{-4}$"></TD>
<TD ALIGN="CENTER">
<!-- MATH
 $4.6 \cdot 10^{-5}$
 -->
<IMG
 WIDTH="75" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
 SRC="img687.gif"
 ALT="$4.6 \cdot 10^{-5}$"></TD>
<TD ALIGN="CENTER">
<!-- MATH
 $1.9 \cdot 10^{-5}$
 -->
<IMG
 WIDTH="75" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
 SRC="img688.gif"
 ALT="$1.9 \cdot 10^{-5}$"></TD>
<TD ALIGN="CENTER">
<!-- MATH
 $1.2 \cdot 10^{-7}$
 -->
<IMG
 WIDTH="75" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
 SRC="img500.gif"
 ALT="$1.2\cdot10^{-7}$"></TD>
</TR>
<TR><TD ALIGN="CENTER">3</TD>
<TD ALIGN="CENTER"><B>1010.</B></TD>
<TD ALIGN="CENTER">
<!-- MATH
 $1.7 \cdot 10^{-4}$
 -->
<IMG
 WIDTH="75" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
 SRC="img686.gif"
 ALT="$1.7 \cdot 10^{-4}$"></TD>
<TD ALIGN="CENTER">
<!-- MATH
 $1.0 \cdot 10^{-4}$
 -->
<IMG
 WIDTH="75" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
 SRC="img689.gif"
 ALT="$1.0 \cdot 10^{-4}$"></TD>
<TD ALIGN="CENTER">
<!-- MATH
 $1.9 \cdot 10^{-5}$
 -->
<IMG
 WIDTH="75" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
 SRC="img688.gif"
 ALT="$1.9 \cdot 10^{-5}$"></TD>
<TD ALIGN="CENTER">
<!-- MATH
 $1.1 \cdot 10^{-7}$
 -->
<IMG
 WIDTH="75" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
 SRC="img690.gif"
 ALT="$1.1 \cdot 10^{-7}$"></TD>
</TR>
</TABLE>
</DIV>

<P>
This code fragment cannot be adapted to use xSBGV (or xHBGV),
because xSBGV does not return a conventional Cholesky factor in <B><I>B</I></B>,
but rather a ``split'' Cholesky factorization (performed by xPBSTF).

<P>
<DT>2.
<DD>Problem types 2 and 3 have the same error bounds. We illustrate
only type 2.
<A NAME="11940"></A>
<A NAME="11941"></A>

<P>
 
<P>
<PRE>
      EPSMCH = SLAMCH( 'E' )
*     Solve the eigenproblem A*B - lambda I (ITYPE = 2)
      ITYPE = 2
*     Compute the norms of A and B
      ANORM = SLANSY( '1', UPLO, N, A, LDA, WORK )
      BNORM = SLANSY( '1', UPLO, N, B, LDB, WORK )
*     The eigenvalues are returned in W
*     The eigenvectors are returned in A
      CALL SSYGV( ITYPE, 'V', UPLO, N, A, LDA, B, LDB, W, WORK,
     $            LWORK, INFO )
      IF( INFO.GT.0 .AND. INFO.LE.N ) THEN
         PRINT *,'SSYGV did not converge'
      ELSE IF( INFO.GT.N ) THEN
         PRINT *,'B not positive definite'
      ELSE IF ( N.GT.0 ) THEN
*        Get reciprocal condition number RCONDB of Cholesky factor of B
         CALL STRCON( '1', UPLO, 'N', N, B, LDB, RCONDB, WORK, IWORK,
     $                INFO )
         RCONDB = MAX( RCONDB, EPSMCH )
         CALL SDISNA( 'Eigenvectors', N, N, W, RCONDZ, INFO )
         DO 10 I = 1, N
            EERRBD(I) = ( ANORM * BNORM ) * EPSMCH +
     $                  ( EPSMCH / RCONDB**2 ) * ABS( W(I) )
            ZERRBD(I) = ( EPSMCH / RCONDB ) * ( ( ANORM * BNORM ) /
     $                    RCONDZ(I) + 1.0 / RCONDB )
10       CONTINUE
      END IF
</PRE>
<A NAME="11945"></A>

<P>
For the same <B><I>A</I></B> and <B><I>B</I></B> as above, the approximate eigenvalues,
approximate error bounds, and true errors are

<P>
<DIV ALIGN="CENTER">
<TABLE CELLPADDING=3 BORDER="1">
<TR><TD ALIGN="CENTER"><B><I>i</I></B></TD>
<TD ALIGN="CENTER"><IMG
 WIDTH="20" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
 SRC="img523.gif"
 ALT="$\lambda_i$"></TD>
<TD ALIGN="CENTER"><TT> EERRBD</TT><B>(<I>i</I>)</B></TD>
<TD ALIGN="CENTER">true 
<!-- MATH
 $| \hat{\lambda}_i - \lambda_i |$
 -->
<IMG
 WIDTH="67" HEIGHT="41" ALIGN="MIDDLE" BORDER="0"
 SRC="img531.gif"
 ALT="$\vert \hat{\lambda}_i - \lambda_i \vert$"></TD>
<TD ALIGN="CENTER"><TT> ZERRBD</TT><B>(<I>i</I>)</B></TD>
<TD ALIGN="CENTER">true 
<!-- MATH
 $\theta ( \hat{v}_i , v_i )$
 -->
<IMG
 WIDTH="62" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="img568.gif"
 ALT="$\theta ( \hat{v}_i , v_i )$"></TD>
</TR>
<TR><TD ALIGN="CENTER">1</TD>
<TD ALIGN="CENTER">
<!-- MATH
 $-4.817 \cdot 10^6$
 -->
<IMG
 WIDTH="95" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="img691.gif"
 ALT="$-4.817 \cdot 10^6$"></TD>
<TD ALIGN="CENTER"><B>1.3</B></TD>
<TD ALIGN="CENTER">
<!-- MATH
 $6.0 \cdot 10^{-3}$
 -->
<IMG
 WIDTH="75" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
 SRC="img692.gif"
 ALT="$6.0 \cdot 10^{-3}$"></TD>
<TD ALIGN="CENTER">
<!-- MATH
 $1.7 \cdot 10^{-7}$
 -->
<IMG
 WIDTH="75" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
 SRC="img693.gif"
 ALT="$1.7 \cdot 10^{-7}$"></TD>
<TD ALIGN="CENTER">
<!-- MATH
 $7.0 \cdot 10^{-9}$
 -->
<IMG
 WIDTH="75" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
 SRC="img694.gif"
 ALT="$7.0 \cdot 10^{-9}$"></TD>
</TR>
<TR><TD ALIGN="CENTER">2</TD>
<TD ALIGN="CENTER">
<!-- MATH
 $8.094 \cdot 10^6$
 -->
<IMG
 WIDTH="82" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
 SRC="img695.gif"
 ALT="$8.094 \cdot 10^6$"></TD>
<TD ALIGN="CENTER"><B>1.6</B></TD>
<TD ALIGN="CENTER"><B>1.5</B></TD>
<TD ALIGN="CENTER">
<!-- MATH
 $3.4 \cdot 10^{-7}$
 -->
<IMG
 WIDTH="75" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
 SRC="img696.gif"
 ALT="$3.4 \cdot 10^{-7}$"></TD>
<TD ALIGN="CENTER">
<!-- MATH
 $3.3 \cdot 10^{-8}$
 -->
<IMG
 WIDTH="75" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
 SRC="img697.gif"
 ALT="$3.3 \cdot 10^{-8}$"></TD>
</TR>
<TR><TD ALIGN="CENTER">3</TD>
<TD ALIGN="CENTER">
<!-- MATH
 $1.219 \cdot 10^7$
 -->
<IMG
 WIDTH="82" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
 SRC="img698.gif"
 ALT="$1.219 \cdot 10^7$"></TD>
<TD ALIGN="CENTER"><B>1.9</B></TD>
<TD ALIGN="CENTER"><B>4.5</B></TD>
<TD ALIGN="CENTER">
<!-- MATH
 $3.4 \cdot 10^{-7}$
 -->
<IMG
 WIDTH="75" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
 SRC="img696.gif"
 ALT="$3.4 \cdot 10^{-7}$"></TD>
<TD ALIGN="CENTER">
<!-- MATH
 $4.7 \cdot 10^{-8}$
 -->
<IMG
 WIDTH="75" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
 SRC="img699.gif"
 ALT="$4.7 \cdot 10^{-8}$"></TD>
</TR>
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<I>Susan Blackford</I>
<BR><I>1999-10-01</I>
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